Electromagnetism Ch.7Methods of Math. Physics, Friday 8 April 2011, EJZ

• Inductors and inductance

• Waves and wave equations

• Electromagnetism & Maxwell’s eqns

• Derive EM wave equation and speed of light

Faraday’s Law

Break time…

Waves

( , ) sin( )MD x t D kx t

Wave equation1. Differentiate D/t 2D/t2

2. Differentiate D/x 2D/x2

3. Find the speed from

22

222 2 22 2

2

2

2

Dt T f v

D k Tx

Causes and effects of E

Gauss: E fields diverge from charges

Lorentz force: E fields can move charges

0

dq

E A F = q E

Causes and effects of B

Ampere: B fields curl around currents

Lorentz force: B fields can bend moving charges

0dl B I F = q v x B = IL x B

Changing fields create new fields!

Faraday: Changing magnetic flux induces circulating electric field

lE ddt

d B

Guess what a changing E field induces?

Changing E field creates B field!

Current piles charge onto capacitor

Magnetic field doesn’t stop

Changing electric flux

→“displacement current”

→ magnetic circulation

lB ddt

d E

00AE dE

Partial Maxwell’s equations

Charge → E field Current → B field

0

E dAq

0B dl I

lE ddt

d B lB d

dt

d E

00

FaradayChanging B → E

AmpereChanging E → B

Maxwell eqns → electromagnetic waves

Consider waves traveling in the x direction with frequency f =

and wavelength = /k

E(x,t)=E0 sin (kx-t) and

B(x,t)=B0 sin (kx-t)

Do these solve Faraday and Ampere’s laws?

Faraday + Ampere

lE ddt

d B

dx

dB

dt

dE00

dx

dE

dt

dB

lB ddt

d E

00

dx

dB

dt

dE00dx

dE

dt

dB

Sub in: E=E0 sin (kx-t) and B=B0 sin (kx-t)

Speed of Maxwellian waves?

Faraday: = k E0 Ampere: E0=kB0

Eliminate B0/E0 and solve for v=/k

x10-7 Tm/A = 8.85 x 10-12 C2 N/m2

Maxwell equations → Light

E(x,t)=E0 sin (kx-t) and B(x,t)=B0 sin (kx-t)solve Faraday’s and Ampere’s laws.

Electromagnetic waves in vacuum have speed c and energy/volume = 1/2 0 E2 = B2 /(20 )

Full Maxwell equations inintegral form

0

E dAq

B dA 0

Bdd

dt

E l

0 0 0Ed

d Idt

B l

Integral to differential form

0

E dA

v dA v

q

d

dqq d d

d

Gauss’ Law: apply

Divergence Thm: and the

Definition of charge density: to find the

Differential form:

Integral to differential form

0B dl

v dl v dA

I

dII J dA dA

dA

Ampere’s Law: apply

Curl Thm: and the

Definition of current density: to find the

Differential form:

Integral to differential form

dA

v dl v dA

Bd dd

dt dt

E l B

Faraday’s Law: apply

Curl Thm: to find the

Differential form:

Finish integral to differential form…

0

0

E dA

E=

q

B dA 0

Bdd

dtd

dt

E l

BE

0 0 0Ed

d Idt

B l

Finish integral to differential form…

0

0

d

=

q

E A

E

d 0

0

B A

B

Bdd

dtd

dt

E l

BE

0 0 0

0 0 0

Edd

dtd

dt

B l I

EB J

Maxwell eqns in differential form

0

=

E

0 B

d

dt

BE

0 0 0

d

dt

EB J